 Previously we saw the harmonic oscillator. It's a simple model. The restoring force of a bond or a spring is directly proportional to the displacement. And the frequency of the oscillation, the harmonic frequency, is the same regardless of that displacement. But there is a problem with this model. As you begin to compress a real bond, the restoring force must go up much higher than the harmonic oscillator's jests. And as you extend it, the bond must eventually break. And this model is known as the anharmonic oscillator. Anharmonicity means that the average bond length and the frequency of the vibration actually changes depending on the energy level. Now the mathematics and quantum mechanics of that can be a little complicated, but we don't need to know too much about it. And the main equation that we end up using is simply a fudge factor that maps the harmonic to the anharmonic frequency and vice versa. And that works because at lower energy levels, the harmonic oscillator is actually a very good approximation of the real system. Our starting point is called the Morse potential. We won't go into it mathematically, but let's just look at the shape qualitatively and related to some of the terms. The A term is related to the force constant at the bottom of the potential well. Because the potential energy well here is asymmetrical, the force constant isn't exactly constant, so it applies to this point only. The D E term is the dissociation energy for the bond. The energy required to fully dissociate both atoms. Closer to zero distance, this curve increases dramatically, much more so when the force constant is high compared to the dissociation energy. The final term is the equilibrium bond length, R E, which simply acts to shift the potential curve left and right on this axis. Interestingly, the Morse potential doesn't actually go to infinity at zero distance. But in practice, you're never going to see it on a diagram that's reasonably scaled so that you can see this dissociation energy. So instead, let's just look at the differences between these two potentials. Now, on the Morse potential, you're just going to shoot up very, very rapidly as the bond distance closes in. There's also a clear tail as you begin to approach the dissociation energy. Meanwhile, on the harmonical oscillator, because that is symmetrical, it implies you can get much closer to zero distance for far less energy. And it also implies that the bond should never break because it keeps going up and energy as the bond increases. The remaining difference is in how energy levels work. In the harmonical oscillator, the energy levels are spaced perfectly evenly and the equilibrium or average bond distance remains the same at any energy level. In the anharmonic oscillator, there's a limit. The energy levels begin to bunch up as the quantum number V increases. Anharmonicity has an effect on the spectrum because of this bunching of the energy levels. They're not exactly as predicted by the harmonic oscillator. And it's these spacings that we actually begin to see on a spectrum. So what we really need is some way to convert the energies we see on an anharmonic oscillator in the real world to the harmonic one, where we can actually begin to get useful information from it because it's the simpler model. We begin by looking at the energy of each vibrational level in our harmonic oscillator. This is related to the frequency of that oscillator, omega, and the vibrational quantum number, with the plus half being the zero point energy mentioned in a previous video. The tilde on top means we're working in wave numbers so we don't need to convert to joules or other units in this case. Now we tweak it to make an anharmonic version by adding another copy of the quantum number, making it quadratic, then fudging it with a factor known as the anharmonicity constant. We can add a higher order term, but this is usually good enough for most purposes. This is probably the longest equation we'll meet in this sort of course, but it does crunch down to something simpler later. The aim here is to somehow convert the anharmonic frequencies to harmonic ones and allow us to extract a meaningful force constant from it. We're not going to cover how the maths of that works in these videos, we'll do that as part of a workshop. So for now that's it for infrared absorption, and we'll move on to the next bit of vibrational spectroscopy, the Raman spectrum.